Theorem ragperp 28705

Description: Deduce that two lines are perpendicular from a right angle statement. One direction of theorem 8.13 of [Schwabhauser] p. 59. (Contributed by Thierry Arnoux, 20-Oct-2019.)

Hypotheses

Ref	Expression
isperp.p	⊢ 𝑃 = (Base‘𝐺)
isperp.d	⊢ − = (dist‘𝐺)
isperp.i	⊢ 𝐼 = (Itv‘𝐺)
isperp.l	⊢ 𝐿 = (LineG‘𝐺)
isperp.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
isperp.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
ragperp.b	⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
ragperp.x	⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))
ragperp.u	⊢ (𝜑 → 𝑈 ∈ 𝐴)
ragperp.v	⊢ (𝜑 → 𝑉 ∈ 𝐵)
ragperp.1	⊢ (𝜑 → 𝑈 ≠ 𝑋)
ragperp.2	⊢ (𝜑 → 𝑉 ≠ 𝑋)
ragperp.r	⊢ (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))

Assertion

Ref	Expression
ragperp	⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)

Proof of Theorem ragperp

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isperp.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		isperp.d	. . . 4 ⊢ − = (dist‘𝐺)
3		isperp.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		isperp.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
5		eqid 2733	. . . 4 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
6		isperp.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ TarskiG)
8		ragperp.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐵 ∈ ran 𝐿)
10		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵)
11	1, 4, 3, 7, 9, 10	tglnpt 28537	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝑃)
12		isperp.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
13	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐴 ∈ ran 𝐿)
14		ragperp.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))
15	14	elin1d 4155	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
16	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑋 ∈ 𝐴)
17	1, 4, 3, 7, 13, 16	tglnpt 28537	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑋 ∈ 𝑃)
18		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝐴)
19	1, 4, 3, 7, 13, 18	tglnpt 28537	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝑃)
20		ragperp.v	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
21	20	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ∈ 𝐵)
22	1, 4, 3, 7, 9, 21	tglnpt 28537	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ∈ 𝑃)
23		ragperp.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
24	23	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ∈ 𝐴)
25	1, 4, 3, 7, 13, 24	tglnpt 28537	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ∈ 𝑃)
26		ragperp.r	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))
27	26	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑈𝑋𝑉”⟩ ∈ (∟G‘𝐺))
28		ragperp.1	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≠ 𝑋)
29	28	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ≠ 𝑋)
30	23	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑈 ∈ 𝐴)
31	6	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐺 ∈ TarskiG)
32	17	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ∈ 𝑃)
33	19	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑢 ∈ 𝑃)
34		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → ¬ 𝑋 = 𝑢)
35	34	neqned 2937	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ≠ 𝑢)
36	12	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐴 ∈ ran 𝐿)
37	15	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ∈ 𝐴)
38	18	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑢 ∈ 𝐴)
39	1, 3, 4, 31, 32, 33, 35, 35, 36, 37, 38	tglinethru 28624	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐴 = (𝑋𝐿𝑢))
40	30, 39	eleqtrd 2835	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑈 ∈ (𝑋𝐿𝑢))
41	40	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (¬ 𝑋 = 𝑢 → 𝑈 ∈ (𝑋𝐿𝑢)))
42	41	orrd 863	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑋 = 𝑢 ∨ 𝑈 ∈ (𝑋𝐿𝑢)))
43	42	orcomd 871	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑈 ∈ (𝑋𝐿𝑢) ∨ 𝑋 = 𝑢))
44	1, 2, 3, 4, 5, 7, 25, 17, 22, 19, 27, 29, 43	ragcol 28687	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑢𝑋𝑉”⟩ ∈ (∟G‘𝐺))
45	1, 2, 3, 4, 5, 7, 19, 17, 22, 44	ragcom 28686	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑉𝑋𝑢”⟩ ∈ (∟G‘𝐺))
46		ragperp.2	. . . . . 6 ⊢ (𝜑 → 𝑉 ≠ 𝑋)
47	46	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ≠ 𝑋)
48	20	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑉 ∈ 𝐵)
49	6	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐺 ∈ TarskiG)
50	17	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ∈ 𝑃)
51	11	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑣 ∈ 𝑃)
52		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → ¬ 𝑋 = 𝑣)
53	52	neqned 2937	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ≠ 𝑣)
54	8	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐵 ∈ ran 𝐿)
55	14	elin2d 4156	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
56	55	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ∈ 𝐵)
57	10	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑣 ∈ 𝐵)
58	1, 3, 4, 49, 50, 51, 53, 53, 54, 56, 57	tglinethru 28624	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐵 = (𝑋𝐿𝑣))
59	48, 58	eleqtrd 2835	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑉 ∈ (𝑋𝐿𝑣))
60	59	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (¬ 𝑋 = 𝑣 → 𝑉 ∈ (𝑋𝐿𝑣)))
61	60	orrd 863	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑋 = 𝑣 ∨ 𝑉 ∈ (𝑋𝐿𝑣)))
62	61	orcomd 871	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑉 ∈ (𝑋𝐿𝑣) ∨ 𝑋 = 𝑣))
63	1, 2, 3, 4, 5, 7, 22, 17, 19, 11, 45, 47, 62	ragcol 28687	. . . 4 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑣𝑋𝑢”⟩ ∈ (∟G‘𝐺))
64	1, 2, 3, 4, 5, 7, 11, 17, 19, 63	ragcom 28686	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺))
65	64	ralrimivva 3177	. 2 ⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺))
66	1, 2, 3, 4, 6, 12, 8, 14	isperp2 28703	. 2 ⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ⟨“𝑢𝑋𝑣”⟩ ∈ (∟G‘𝐺)))
67	65, 66	mpbird 257	1 ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049   ∩ cin 3898   class class class wbr 5095  ran crn 5622  ‘cfv 6489  (class class class)co 7355  ⟨“cs3 14759  Basecbs 17130  distcds 17180  TarskiGcstrkg 28415  Itvcitv 28421  LineGclng 28422  pInvGcmir 28640  ∟Gcrag 28681  ⟂Gcperpg 28683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11072  ax-resscn 11073  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-addrcl 11077  ax-mulcl 11078  ax-mulrcl 11079  ax-mulcom 11080  ax-addass 11081  ax-mulass 11082  ax-distr 11083  ax-i2m1 11084  ax-1ne0 11085  ax-1rid 11086  ax-rnegex 11087  ax-rrecex 11088  ax-cnre 11089  ax-pre-lttri 11090  ax-pre-lttrn 11091  ax-pre-ltadd 11092  ax-pre-mulgt0 11093

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-oadd 8398  df-er 8631  df-map 8761  df-pm 8762  df-en 8879  df-dom 8880  df-sdom 8881  df-fin 8882  df-dju 9804  df-card 9842  df-pnf 11158  df-mnf 11159  df-xr 11160  df-ltxr 11161  df-le 11162  df-sub 11356  df-neg 11357  df-nn 12136  df-2 12198  df-3 12199  df-n0 12392  df-xnn0 12465  df-z 12479  df-uz 12743  df-fz 13418  df-fzo 13565  df-hash 14248  df-word 14431  df-concat 14488  df-s1 14514  df-s2 14765  df-s3 14766  df-trkgc 28436  df-trkgb 28437  df-trkgcb 28438  df-trkg 28441  df-cgrg 28499  df-mir 28641  df-rag 28682  df-perpg 28684

This theorem is referenced by:  footexALT  28706  footexlem2  28708  colperpexlem3  28720  mideulem2  28722  lmimid  28782  hypcgrlem1  28787  hypcgrlem2  28788  trgcopyeulem  28793